1. Today: Lab 9ab due after lecture: CEQ Monday: Quizz 11: review Wednesday: Guest lecture – Multivariate Analysis Friday: last lecture: review – Bring questions DEC 8 – 9am FINAL EXAM EN 2007

2. Biology 4605 / 7220Name ________________ Quiz #10a19 November 2012 1.What are the 2 main differences between general linear models and generalized linear models? 2. A generalized linear model links a response variable to one or more explanatory variables Xi according to a link function.

3. Biology 4605 / 7220Name ________________ Quiz #10a19 November 2012 1.What are the 2 main differences between general linear models and generalized linear models? Most common answers: A. Non –normal ε B. ANODEV instead of ANOVA table C. Link function 2. A generalized linear model links a response variable to one or more explanatory variables Xi according to a link function. conceptual implementation

4. GLM, GzLM, GAM A few concepts and ideas

5. GLM Model based statistics – we define the response and the explanatory without worrying about the name of the test

6. GLM t-test ANOVA Simple Linear Regression Multiple Linear Regression ANCOVA GENERAL LINEAR MODELS ε ~ Normal R: lm()

7. GLM An example from Lab 9

8. GLM Do fumigants (treatments) decrease the number of wire worms? #ww = β 0 + β treatment treatment + β row row + β column column treatment  fixed row  random column  random N=25

9. GLM N=25

10. GLM N=25

11. GLM N=25

12. GLM N=25

13. GLM p-value borderline Normality assumption not met

14. GLM N=25 p-value borderline Normality assumption not met n<30 Given that we do not violate the homogeneity assumption, randomizing will likely not change our decision… or will it? Let’s try  p rand = 0.0626 (50 000 randomizations)

15. GLM Parameters: Means with 95% CI Anything wrong with this analysis?

16. GLM Response variable? Counts

17. GzLM Poisson error #ww = e μ + ε μ = β 0 + β treatment treatment + β row row + β column column

18. GzLM Poisson error #ww = e μ + ε μ = β 0 + β treatment treatment + β row row + β column column ALL fits > 0

19. GzLM Poisson error

21. t-test ANOVA Simple Linear Regression Multiple Linear Regression ANCOVA Poisson Binomial Negative Binomial Gamma Multinomial GENERALIZED LINEAR MODELS Inverse Gaussian Exponential GENERAL LINEAR MODELS ε ~ Normal Linear combination of parameters R: lm() R: glm() GzLM

22. #ww = e μ + ε μ = β 0 + β treatment treatment + β row row + β column column Generalized linear models have 3 components: Systematic Random Link function

23. GzLM #ww = e μ + ε μ = β 0 + β treatment treatment + β row row + β column column Generalized linear models have 3 components: Systematic linear predictor Random Link function

24. GzLM #ww = e μ + ε μ = β 0 + β treatment treatment + β row row + β column column Generalized linear models have 3 components: Systematic linear predictor Random probability distribution  poisson error Link function

25. GzLM #ww = e μ + ε μ = β 0 + β treatment treatment + β row row + β column column Generalized linear models have 3 components: Systematic linear predictor Random probability distribution  poisson error Link function log

26. GzLM

27. GLM An example from Lab 6

28. GLM Do movements of juvenile cod depend on time of day? distance = β 0 + β period period period  categorical

29. GLM

30. Anything wrong with this analysis?

31. GAM

32. t-test ANOVA Simple Linear Regression Multiple Linear Regression ANCOVA Poisson Binomial Negative Binomial Gamma Multinomial GENERALIZED LINEAR MODELS Inverse Gaussian Exponential Non-linear effect of covariates GENERALIZED ADDITIVE MODELS GENERAL LINEAR MODELS ε ~ Normal Linear combination of parameters R: lm() R: glm() R: gam() GAM

33. Generalized case of generalized linear models where the systematic component is not necessarily linear distance ~ s(period) y ~ s(x 1 ) + s(x 2 ) + x 3 + …. s: smooth function Spline functions are concerned with good approximation of functions over the whole of a region, and behave in a stable manner

34. GAM Smoothing - concept

35. Degree of smoothness -+ GAM How much smoothing?

36. GAM

37. t-test ANOVA Simple Linear Regression Multiple Linear Regression ANCOVA GENERAL LINEAR MODELS ε ~ Normal R: lm()

38. t-test ANOVA Simple Linear Regression Multiple Linear Regression ANCOVA Poisson Binomial Negative Binomial Gamma Multinomial GENERALIZED LINEAR MODELS Inverse Gaussian Exponential GENERAL LINEAR MODELS ε ~ Normal Linear combination of parameters R: lm() R: glm() Non-normal ε Link function

39. t-test ANOVA Simple Linear Regression Multiple Linear Regression ANCOVA Poisson Binomial Negative Binomial Gamma Multinomial GENERALIZED LINEAR MODELS Inverse Gaussian Exponential Non-linear effect of covariates GENERALIZED ADDITIVE MODELS GENERAL LINEAR MODELS ε ~ Normal Linear combination of parameters R: lm() R: glm() R: gam() Linear predictor involves sums of smooth functions of covariates